Question: Determine how many solutions exist for the system of equations. ${3x+y = 10}$ ${6x+3y = 27}$
Solution: Convert both equations to slope-intercept form: ${3x+y = 10}$ $3x{-3x} + y = 10{-3x}$ $y = 10-3x$ ${y = -3x+10}$ ${6x+3y = 27}$ $6x{-6x} + 3y = 27{-6x}$ $3y = 27-6x$ $y = 9-2x$ ${y = -2x+9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -3x+10}$ ${y = -2x+9}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.